# sheaf

## 1 Presheaves

Let $X$ be a topological space and let $\mathcal{A}$ be a category. A presheaf on $X$ with values in $\mathcal{A}$ is a contravariant functor $F$ from the category $\mathcal{B}$ whose objects are open sets in $X$ and whose morphisms are inclusion mappings of open sets of $X$, to the category $\mathcal{A}$.

As this definition may be less than helpful to many readers, we offer the following equivalent (but longer) definition. A presheaf $F$ on $X$ consists of the following data:

1. 1.

An object $F(U)$ in $\mathcal{A}$, for each open set $U\subset X$

2. 2.

A morphism $\operatorname{res}_{V,U}\colon F(V)\to F(U)$ for each pair of open sets $U\subset V$ in $X$ (called the restriction morphism), such that:

1. (a)

For every open set $U\subset X$, the morphism $\operatorname{res}_{U,U}$ is the identity morphism.

2. (b)

For any open sets $U\subset V\subset W$ in $X$, the diagram